Transcript 14-1 - Militant Grammarian

14-1Graphs
14-1
GraphsofofSine
Sineand
andCosine
Cosine
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
14-1 Graphs of Sine and Cosine
Warm Up
Evaluate.
1.
0.5
2.
3.
0.5
4.
0
Find the measure of the reference angle
for each given angle.
5. 145° 35°
Holt Algebra 2
5. 317° 43°
14-1 Graphs of Sine and Cosine
Objective
Recognize and graph periodic and
trigonometric functions.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Vocabulary
periodic function
cycle
period
amplitude
frequency
phase shift
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Periodic functions are functions that repeat
exactly in regular intervals called cycles. The
length of the cycle is called its period. Examine
the graphs of the periodic function and
nonperiodic function below. Notice that a cycle
may begin at any point on the graph of a
function.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Example 1A: Identifying Periodic Functions
Identify whether each function is periodic. If
the function is periodic, give the period.
The pattern repeats
exactly, so the function is
periodic. Identify the
period by using the start
and finish of one cycle.
This function is periodic
with a period of .
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Example 1B: Identifying Periodic Functions
Identify whether each function is periodic. If
the function is periodic, give the period.
Although there is some
symmetry, the pattern does
not repeat exactly. This
function is not periodic.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Check It Out! Example 1
Identify whether each function is periodic. If
the function is periodic, give the period.
a.
b.
not periodic
Holt Algebra 2
periodic; 3
14-1 Graphs of Sine and Cosine
The trigonometric functions that you studied in
Chapter 13 are periodic. You can graph the function
f(x) = sin x on the coordinate plane by using yvalues from points on the unit circle where the
independent variable x represents the angle θ in
standard position.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Similarly, the function f(x) = cos x can be graphed
on the coordinate plane by using x-values from
points on the unit circle.
The amplitude of sine and cosine functions is half
of the difference between the maximum and
minimum values of the function. The amplitude is
always positive.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Holt Algebra 2
14-1 Graphs of Sine and Cosine
You can use the parent functions to graph
transformations y = a sin bx and y = a cos bx.
Recall that a indicates a vertical stretch (|a|>1) or
compression (0 < |a| < 1), which changes the
amplitude. If a is less than 0, the graph is
reflected across the x-axis. The value of b
indicates a horizontal stretch or compression,
which changes the period.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Example 2: Stretching or Compressing Functions
Sine and Cosine Functions
Using f(x) = sin x as a guide, graph the
function g(x) =
Identify the
amplitude and period.
Step 1 Identify the amplitude and period.
Holt Algebra 2
Because
the amplitude is
Because
the period is
14-1 Graphs of Sine and Cosine
Example 2 Continued
Step 2 Graph.
The curve is vertically
compressed by a factor of
horizontally stretched by a
factor of 2.
The parent function f has x-intercepts at
multiplies of  and g has x-intercepts at
multiplies of 4 .
The maximum value of g is
value is
.
Holt Algebra 2
, and the minimum
14-1 Graphs of Sine and Cosine
Check It Out! Example 2
Using f(x) = cos x as a guide, graph the
function h(x) =
and period.
Identify the amplitude
Step 1 Identify the amplitude and period.
Because
the amplitude is
Because b = 2, the period is
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Check It Out! Example 2 Continued
Step 2 Graph.
The curve is vertically
compressed by a factor of
and horizontally compressed
by a factor of 2.
The parent function f has x-intercepts at
multiplies of  and h has x-intercepts at
multiplies of .
The maximum value of h is
value is
.
Holt Algebra 2
, and the minimum
14-1 Graphs of Sine and Cosine
Sine and cosine functions can be used to model
real-world phenomena, such as sound waves.
Different sounds create different waves. One way
to distinguish sounds is to measure frequency.
Frequency is the number of cycles in a given unit
of time, so it is the reciprocal of the period of a
function.
Hertz (Hz) is the standard measure of frequency
and represents one cycle per second. For example,
the sound wave made by a tuning fork for middle A
has a frequency of 440 Hz. This means that the
wave repeats 440 times in 1 second.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Example 3: Sound Application
Use a sine function to graph a sound wave
with a period of 0.002 s and an amplitude
of 3 cm. Find the frequency in hertz for this
sound wave.
period
Use a horizontal scale
amplitude
where one unit represents
0.002 s to complete one
full cycle. The maximum
and minimum values are
given by the amplitude.
The frequency of the sound wave is 500 Hz.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Check It Out! Example 3
Use a sine function to graph a sound wave
with a period of 0.004 s and an amplitude
of 3 cm. Find the frequency in hertz for this
sound wave.
Use a horizontal scale
where one unit represents
0.004 s to complete one
full cycle. The maximum
and minimum values are
given by the amplitude.
period
amplitude
The frequency of the sound wave is 250 Hz.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Sine and cosine can also be translated as y
= sin(x – h) + k and y = cos(x – h) + k.
Recall that a vertical translation by k units
moves the graph up (k > 0) or down
(k < 0).
A phase shift is a horizontal translation of a
periodic function. A phase shift of h units moves
the graph left (h < 0) or right (h > 0).
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Example 4: Identifying Phase Shifts for Sine and
Cosine Functions
Using f(x) = sin x as a guide, graph g(x) =
g(x) = sin
Identify the x-intercepts and phase shift.
Step 1 Identify the amplitude and period.
Amplitude is |a| = |1| = 1.
The period is
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Example 4 Continued
Step 2 Identify the phase shift.
Identify h.
Because h =
the phase shift is
radians to the right.
All x-intercepts, maxima, and minima
of f(x) are shifted units to the right.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Example 4 Continued
Step 3 Identify the x-intercepts.
The first x-intercept occurs at .
Because cos x has two x-intercepts in
each period of 2, the x-intercepts
occur at + n, where n is an integer.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Example 4 Continued
Step 4 Identify the maximum and minimum values.
The maximum and minimum values occur
between the x-intercepts. The maxima
occur at
+ 2n and have a value of 1.
The minima occur at + 2n and have a
value of –1.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Example 4 Continued
Step 5 Graph using all the information about the
function.
sin x
sin
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Check It Out! Example 4
Using f(x) = cos x as a guide, graph
g(x) = cos(x – ). Identify the x-intercepts
and phase shift.
Step 1 Identify the amplitude and period.
Amplitude is |a| = |1| = 1.
The period is
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Check It Out! Example 4 Continued
Step 2 Identify the phase shift.
x –  = x – (–)
Identify h.
Because h = –, the phase shift is 
radians to the right.
All x-intercepts, maxima, and minima
of f(x) are shifted  units to the right.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Check It Out! Example 4 Continued
Step 3 Identify the x-intercepts.
The first x-intercept occurs at .
Because sin x has two x-intercepts in
each period of 2, the x-intercepts
occur at + n, where n is an integer.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Check It Out! Example 4 Continued
Step 4 Identify the maximum and minimum values.
The maximum and minimum values occur
between the x-intercepts. The maxima
occur at  + 2n and have a value of 1. The
minima occur at 2n and have a value of –1.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Check It Out! Example 4 Continued
Step 5 Graph using all the information about the
function.
y
cos x
–

x
cos (x–)
Holt Algebra 2
14-1 Graphs of Sine and Cosine
You can combine the transformations of
trigonometric functions. Use the values of a, b,
h, and k to identify the important features of a
sine or cosine function.
Amplitude
Phase shift
y = asinb(x – h) + k
Period
Holt Algebra 2
Vertical shift
14-1 Graphs of Sine and Cosine
Example 5: Employment Application
The number of people, in thousands, employed
in a resort town can be modeled by,
where x is the
month of the year.
A. Graph the number of people employed in the
town for one complete period.
a = 1.5, b =
k = 5.2
, h = –2,
Step 1 Identify the important features of the graph.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Example 4 Continued
Amplitude: 1.5
Period:
The period is equal to 12 months or 1 full year.
Phase shift: 2 months left
Vertical shift: 5.2
Maxima: 5.2 + 1.5 = 6.7 at 1
Minima: 5.2 – 1.5 = 3.7 at 7
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Example 4 Continued
Step 2 Graph using all the information about
the function.
B. What is the maximum number of people
employed?
The maximum number of people employed is
1000(5.2 + 1.5) = 6700.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Check It Out! Example 5
What if…? Suppose that the height H of a
Ferris wheel can be modeled by,
, where t is the
time in seconds.
a. Graph the height of a cabin for two complete
periods.
H(t) = –16cos
+ 24
a = –16, b =
, k = 24
Step 1 Identify the important features of the
graph.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Check It Out! Example 5 Continued
Amplitude: –16
Period:
The period is equal to the time required for one
full rotation.
Vertical shift: 24
Maxima: 24 + 16 = 40
Minima: 24 – 16 = 8
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Height (ft)
Check It Out! Example 5 Continued
Time (min)
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Lesson Quiz: Part I
1. Using f(x) = cos x as a guide, graph
g(x) = 1.5 cos 2x.
Holt Algebra 2
14-1 Graphs of Sine and Cosine
Lesson Quiz: Part II
Suppose that the height, in feet, above ground
of one of the cabins of a Ferris wheel at t
minutes is modeled by
2. Graph the height of the cabin for two complete
revolutions.
3. What is the radius of this
Ferris wheel? 30 ft
Holt Algebra 2